Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions

TL;DR

This paper explores the relationships between fractional positive semidefinite rank, projective rank, and orthogonal representations of graphs, revealing new connections and directions with implications for quantum theory.

Contribution

It introduces the fractional minimum positive semidefinite rank and $r$-fold orthogonal representations, linking these concepts to projective rank and quantum theory.

Findings

Projective rank equals fractional minimum positive semidefinite rank of a graph's complement.

Introduces $r$-fold versions of positive semidefinite rank and orthogonal representations.

Discusses connections to quantum theory, including Tsirelson's problem.

Abstract

Fractional minimum positive semidefinite rank is defined from $r$-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An $r$-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. This paper also introduces $r$-fold orthogonal representations of graphs and formalizes the understanding of projective rank as fractional orthogonal rank. Connections of these concepts to quantum theory, including Tsirelson's problem, are discussed.